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# P1.T1.32.1 Practice Question - Information Ratio

#### gprisby

##### Active Member
Hello,

Looking for a little clarification on this question

I understand IR to be:

Rp - Rb / s.d.(Rp - Rb) <-----Denominator also known as TE

also defined as:

Alpha / TE

Now to the question:

32.1 Make the following assumptions:

 Riskfree rate is 3%
 The benchmark is the market (i.e., CAPM) and the benchmark return was 8%
 Portfolio beta is 1.2
 Portfolio return was 10%
 Tracking error was 10%
 Minimum acceptable return (MAR) was 2%
 Downside deviation was 5%

What is (was) the information ratio?

a) 0.10
b) 0.20
c) 0.30
d) 0.40

Alpha = 10% - (1.2 beta * 5% ERP) - 3% riskfree rate = 1%.

IR = alpha/TE = 1%/10% = 0.10

... please note that (B) is tempting because alpha of 10% - 8% is tempting. However, that is

active return not residual return (alpha).

Question:

I guess I don't understand active return vs. residual return? I was really confident it was simply (you also noted this might trick some folks... guess you got me lol):

10% - 8% / 10% = .20

Isn't that the equation for IR?

Last edited by a moderator:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gprisby This (older) question suffered the imprecision/confusion created by conflicting authors in the syllabus. As noted here (https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ ) we subsequently got GARP to settle on "ratio consistency" with respect to IR. As a modern question, this 32.1 has a problem: instead of just "tracking error was 10%," it should (could) give the following assumptions:
• Tracking error (aka, active risk) was x%; σ(P-B) = x%
• Standard deviation of the alpha (aka, residual risk) = 10%.
Then the solution could be either, depending on the question:
• active IR = 2%/x%, or
• residual IR = 1%/10%
Valid are ratio consistent active/active or residual/residual. The problem with "alpha/TE" is that alpha is not really ambiguous (alpha should be residual return) but "tracking error" connotes residual risk but should be defined because it can vary.

Re: I guess I don't understand active return vs. residual return?
• active return = r(p) - r(b) = 10% -8% =+2.0%; this is the out-performance but 1% is due to beta exposure
• residual return = r(p) - [3% + 1.2*(8%-3%)] = +1.0%; i.e., alpha is the excess return that is uncorrelated to common factors so credit for beta is subtracted. When it's the single-factor CAPM beta (rather than many factors), this is the "Jensen's alpha" instance of general alpha. I hope that helps!

#### gprisby

##### Active Member
Thanks David, that helps!

-Greg